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Chapter 4: Problem 10

A student incorrectly simplified \(\frac{6^{5}}{3^{2}}\) as \(2^{3}\). Give the correct answer

Short Answer

Expert verified
The correct simplification is \(2^{5} \times 3^{3}\).

Step by step solution

01

Understand the Incorrect Simplification

The student simplified \(\frac{6^{5}}{3^{2}}\) incorrectly as \(2^{3}\). First, recognize that the student didn't consider the proper base and exponent rules.
02

Simplify the Fraction Calculation

Rewrite \(6^{5}\) and \(3^{2}\) in terms of their prime factors. \(6 = 2 \times 3\), so \(6^{5} = (2 \times 3)^{5} = 2^{5} \times 3^{5}\). Therefore, the expression is \(\frac{2^{5} \times 3^{5}}{3^{2}}\).
03

Cancel Common Factors

Cancel the common factor of \(3^{2}\) in the numerator and denominator: \(\frac{2^{5} \times 3^{5}}{3^{2}} = 2^{5} \times 3^{5-2} = 2^{5} \times 3^{3}\).
04

Write the Final Simplified Answer

The simplified form of \(\frac{6^{5}}{3^{2}}\) is \(2^{5} \times 3^{3}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

simplifying expressions
Simplifying expressions is the process of reducing complex algebraic expressions into simpler forms. This involves combining like terms and applying mathematical operations such as addition, multiplication, and division.
In the given exercise, we had the term \(\frac{6^{5}}{3^{2}}\). Instead of simplifying it directly, we chose to break it down using prime factorization. This allows us to manage and manipulate smaller, simpler numbers which are easier to work with.
When dealing with expressions, remember to:
  • Identify all like terms
  • Use the appropriate exponent rules
  • Always double-check your work to avoid mistakes
The goal is to create a form where all possible simplifications are done, leaving no further reduction possible.
prime factorization
Prime factorization is the process of breaking down a number into its prime factors. Prime factors are the prime numbers that multiply together to yield the original number.
For instance, the number 6 can be written as \(6 = 2 \times 3\). In our exercise, we took \(6^{5}\) and expressed it in terms of its prime factors: \( (2 \times 3)^{5} = 2^{5} \times 3^{5} \).
Prime factorization is helpful because it makes it easier to simplify expressions by exposed common factors, which can then be canceled out. This is seen when we re-wrote \(6^{5}\) in its prime factor form and manipulated it to make the simplification process easier.
cancelling common factors
Cancelling common factors involves eliminating identical factors present in both the numerator and denominator of a fraction. This helps in simplifying the expression to its simplest form.
In our solved problem, we started with the expression \(\frac{2^{5} \times 3^{5}}{3^{2}}\). We noticed that \(3^{2}\) appeared both in the numerator and the denominator. By applying the rule \(a^{m}/a^{n} = a^{m-n} \), we simplified \(3^{5}/3^{2} \) down to \(3^{3} \).
After cancelling the common factor, our result was \( 2^{5} \times 3^{3} \).
Remember, cancelling is vital:
  • Only cancel common factors that appear in both the numerator and the denominator
  • Perform the cancellation correctly to prevent errors
  • Always recheck simplified results
Cancelling common factors is a powerful tool to easily simplify complex algebraic expressions.

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Most popular questions from this chapter

Let \(f(x)=x^{2}-9, g(x)=2 x,\) and \(h(x)=x-3 .\) Find each of the following. $$ (f g)(0) $$

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ 3^{5} \cdot 3^{-6} $$

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ \frac{m^{2} n^{-8}}{m^{-6} n^{3}} $$

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ \frac{\left(m^{4}\right)^{-1}}{9 m^{3}} $$

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ a^{-3} a^{2} a^{-4} $$
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